CN110794459A - Fitting method for seabed near fault seismic oscillation - Google Patents

Fitting method for seabed near fault seismic oscillation Download PDF

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CN110794459A
CN110794459A CN201911113650.6A CN201911113650A CN110794459A CN 110794459 A CN110794459 A CN 110794459A CN 201911113650 A CN201911113650 A CN 201911113650A CN 110794459 A CN110794459 A CN 110794459A
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张超
卢建斌
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Fuzhou University
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Abstract

The invention relates to a fitting method of seabed near fault seismic oscillation, which comprises the following steps: step S1: calculating a seismic motion transfer function of the offshore site; step S2: fitting high-frequency components of the seabed seismic motion to obtain high-frequency acceleration time-range components of the near fault seismic motion in the near sea area; step S3: fitting low-frequency pulse components of the seismic motion of the sea bottom to obtain a low-frequency pulse type acceleration time range of the seismic motion; step S4: and superposing the high-frequency acceleration time-course component and the translated low-frequency pulse type acceleration time-course to obtain the seabed near fault earthquake motion. The method can really solve the bottleneck that the seismic analysis of the existing cross-sea structure lacks seismic record, provides more accurate and reasonable seismic input for the seismic performance analysis of the seabed near fault structure, and is beneficial to promoting and improving the seismic performance evaluation of the sea structure.

Description

Fitting method for seabed near fault seismic oscillation
Technical Field
The invention relates to the field of near fault seismic oscillation evaluation, in particular to a fitting method of seabed near fault seismic oscillation.
Background
Studies of the seismic behavior of near faults have shown that near fault seismic has two significant features that are distinguished from far-field seismic: one is that the speed time interval contains long-period, large-peak pulses; another is that the displacement time course may contain step-type pulses caused by permanent ground displacement. Thus, structures located near faults can produce more severe seismic damage.
Because the existing near-fault seismic motion record is very limited at present, in order to solve the problem that the near-fault seismic motion record is very limited, a relevant student provides a fitting method of the near-fault seismic motion, and the basic thought is as follows: the acceleration time interval of the high frequency component is superimposed with the pulse component of the low frequency component. The artificial near fault seismic motion fitting method enables engineers to more accurately evaluate the seismic performance of the building structure near the near fault, and avoids danger underestimation caused by evaluating the seismic performance only by far-field seismic motion in the past.
However, the existing near-fault seismic motion fitting method is a fitting method proposed based on the near-fault seismic motion record of the land field. Due to the presence of the sea water layer, the seismic propagation law in the sea area environment is significantly different from that of land fields: (1) the sea water layer can directly influence the propagation of seismic waves in an offshore site; (2) the sea water layer can also increase the saturation degree of a soil layer of an offshore site and the pore water pressure, and further the site amplification effect of earthquake motion is influenced. Therefore, the current near-fault seismic motion fitting method aiming at the land field cannot be applied to fitting seabed near-fault seismic motion.
With the rapid development of coastal economy in China, the requirements for constructing life line projects such as large-span traffic projects, large-scale ocean projects and the like near earthquake fault zones or crossing fault zones in sea areas and coastal areas are increasing. For example, a newly built marine bridge is seriously affected by active faults. On the other hand, the existing submarine near fault seismic motion records are few, and the requirements of the existing ocean engineering seismic analysis cannot be met.
At present, no fitting method suitable for seabed near fault seismic motion exists, so that a fitting method suitable for seabed near fault seismic motion needs to be provided urgently, so that a proper seismic motion input is provided for seismic analysis of a sea-crossing structure, and the seismic safety of the sea-crossing structure positioned near a fault is improved.
Disclosure of Invention
In view of the above, the present invention provides a method for fitting a seabed near fault seismic motion, which provides a more accurate and reasonable seismic motion input for the seismic performance analysis of a seabed near fault structure.
The invention is realized by adopting the following scheme: a fitting method of seabed near fault seismic motion comprises the following steps:
step S1: calculating a seismic motion transfer function H (omega) of the offshore site;
step S2: fitting the high-frequency component of the seabed seismic motion to obtain the high-frequency acceleration time-course component a of the near fault seismic motion in the near sea area3(t)；
Step S3: fitting the low-frequency pulse component of the ocean bottom seismic oscillation to obtain the low-frequency pulse type acceleration time course a of the seismic oscillation5(t)；
Step S4: the high-frequency acceleration time course component a3(t) and a low-frequency pulse type acceleration time course a5And (t) superposing to obtain the seabed near fault seismic oscillation.
Further, the step S1 specifically includes the following steps:
step S11: solving dynamic stiffness matrix [ S ] of bedrockR]；
Step S12: dynamic stiffness matrix [ S ] of sea bottom soil layerL]；
Step S13: calculating dynamic stiffness matrix [ S ] of sea water layerW]；
Step S14: dynamic stiffness matrix through sea water layerW]Dynamic stiffness matrix of seabed soil layer SL]And bedrock dynamic stiffness matrix SR]Combining to obtain a dynamic stiffness matrix of the seabed field; and determining a field dynamic balance equation through the dynamic stiffness matrix of the field, the field displacement vector and the field load vector, and solving the equation to obtain the ratio of the bottom displacement of the seabed layer to the displacement at the free surface of the bedrock, namely the offshore field transfer function H (omega).
Further, the step S11 specifically includes the following steps:
calculating the propagation velocity of seismic waves in the bedrock according to the parameter value of the bedrock in the field;
the propagation velocity of the S-wave in the bedrock is expressed as:
Figure BDA0002273461360000031
wherein G is the shear modulus of the bedrock, and rho is the density of the bedrock;
the propagation velocity of the P-wave in the bedrock is expressed as:
Figure BDA0002273461360000032
in the formula, G is the shear modulus of the bedrock, rho is the density of the bedrock, and upsilon is the Poisson ratio of the bedrock;
the S-wave velocity calculation formula considering the hysteresis damping is as follows:
Figure BDA0002273461360000033
in the formula, VpIs the S wave velocity, ξ is the damping ratio;
the P wave velocity calculation formula considering the hysteresis damping is as follows:
Figure BDA0002273461360000034
in the formula, VpP wave velocity, ξ damping ratio;
providing an incident angle of a P wave in the bedrock, and calculating a phase velocity through a calculation formula of the phase velocity; determining phase velocity, and sequentially deducing the incident angles of different seismic waves in other media through a calculation formula of the phase velocity;
the phase velocity is calculated as follows:
wherein c is the phase velocity;
Figure BDA0002273461360000042
and
Figure BDA0002273461360000043
the wave velocities of the P wave and the S wave considering the hysteresis damping are respectively considered; lxAnd mxCosine values of incident angles of the P wave and the S wave respectively; superscripts W, L and R represent the sea water layer, the seabed soil layer and the bedrock, respectively;
obtaining a dynamic stiffness matrix of the bedrock by utilizing the phase velocity, the seismic wave propagation velocity in the bedrock and the parameter value of the bedrock;
the dynamic stiffness matrix of the bedrock under the out-of-plane SH wave action is as follows:
in the formula, GRIs bedrock shear modulus, ξRIs the damping ratio of the bedrock, tRThe tangent value of the incident angle of the out-of-plane SH wave at the bedrock, omega is angular frequency, and c is phase velocity;
the dynamic stiffness matrix of the bedrock under the action of the P wave and the SV wave in the plane is as follows:
Figure BDA0002273461360000045
wherein,
Figure BDA0002273461360000046
Figure BDA0002273461360000047
Figure BDA0002273461360000052
in the formula,
Figure BDA0002273461360000053
shear modulus of bedrock, t 'taking into account hysteresis damping'RIs tangent value, s 'of incident angle of SV wave at bedrock in plane'RThe tangent value of the incident angle of the P wave at the bedrock in the plane is shown, omega is the angular frequency, and c is the phase velocity.
Further, the step S12 specifically includes the following steps:
calculating the Poisson ratio of the water-containing soil body according to the saturation of the seabed soil layer of the offshore site, wherein the Poisson ratio expression of the water-containing soil body is as follows:
Figure BDA0002273461360000054
in the formula, upsilon' and G are respectively Poisson ratio and shear modulus of a soil framework, α and M are respectively coefficients related to compressibility of soil particles and pore fluid, and porosity n and saturation S of the soil body are measuredrCalculating;
calculating the P wave propagation speed of the seabed soil layer by using the Poisson ratio of the water-containing soil body, wherein the P wave propagation speed of the seabed soil layer has the following expression:
Figure BDA0002273461360000055
wherein G is the shear modulus of the soil skeleton, α and M are coefficients related to the compressibility of soil particles and pore fluid respectively, lambda is the Lame constant of the soil skeleton, and rho is (1-n) rhos+nρfDensity of the hydrous soil body, psAnd ρfDensity of the soil particles and pore fluid, respectively;
calculating the propagation speed of S wave in the seabed soil layer according to the parameter value of the seabed soil layer in the field, wherein the S wave speed calculation formula is expressed as follows:
Figure BDA0002273461360000061
in the formula, G is the shear modulus of the soil framework, and rho is the density of the water-containing soil body;
the S-wave velocity calculation formula considering the hysteresis damping is as follows:
Figure BDA0002273461360000062
in the formula, VpIs the S wave velocity, ξ is the damping ratio;
the P wave velocity calculation formula considering the hysteresis damping is as follows:
Figure BDA0002273461360000063
in the formula, VpP wave velocity, ξ damping ratio;
calculating respective incidence angles through a phase velocity calculation formula; the phase velocity is calculated as follows:
wherein c is the phase velocity;and
Figure BDA0002273461360000066
the wave velocities of the P wave and the S wave considering the hysteresis damping are respectively considered; lxAnd mxCosine values of incident angles of the P wave and the S wave respectively; superscripts W, L and R represent the sea water layer, the seabed soil layer and the bedrock, respectively;
obtaining a dynamic stiffness matrix of the seabed soil layer by utilizing the incidence angle of seismic waves in the seabed soil layer and the parameter value of the seabed soil layer; the dynamic stiffness matrix of a certain soil layer is as follows:
wherein each k isLRelating to angular frequency, phase velocity, seismic wave incidence angle and field parameters;
and combining the dynamic stiffness matrixes of all soil layers to obtain a dynamic stiffness matrix of the whole soil layer of the sea bottom.
Further, the step S13 specifically includes the following steps: calculating the P wave propagation speed in the sea water layer according to the parameter value of the sea water layer, wherein the P wave propagation speed expression in the sea water layer is as follows:
Figure BDA0002273461360000071
in the formula, K is the volume modulus of the seawater, and rho is the density of the seawater;
by the formula
Figure BDA0002273461360000072
Calculating the incident angle of the sea water layer P wave by the calculated phase velocity, wherein c is the phase velocity;
Figure BDA0002273461360000073
and
Figure BDA0002273461360000074
the wave velocities of the P wave and the S wave considering the hysteresis damping are respectively considered; lxAnd mxCosine values of incident angles of the P wave and the S wave respectively; superscripts W, L and R represent the sea water layer, the seabed soil layer and the bedrock, respectively;
and combining the depth of the sea water layer and the damping ratio of the sea water layer to obtain a dynamic stiffness matrix of the sea water layer, which is shown in a formula (20).
Wherein each k isWAnd the parameters are related to angular frequency, phase velocity, seismic wave incidence angle, seismic wave propagation velocity, sea water layer shear modulus, sea water layer thickness, sea water layer damping ratio and the like.
Further, the step S14 specifically includes the following steps:
the field dynamic balance equation under the combined action of the P wave and the SV wave in the plane is expressed as follows:
[SP-SV]{uP-SV}＝{PP-SV} (21.1)
wherein [ S ]P-SV]Is a dynamic stiffness matrix of the in-plane direction of the seafloor field, { u }P-SVThe displacement of the field under the combined action of P wave and SV wave, and the displacement of the P wave and SV waveP-SVThe P wave and the SV wave are jointly acted to form a load vector;
the field dynamic equilibrium equation under the out-of-plane SH wave action is expressed as:
[SSH]{uSH}＝{PSH} (21.2)
wherein [ S ]SH]Is a dynamic stiffness matrix of the out-of-plane direction of the seabed ground, { u }SHThe site displacement under the action of SH wave, { P }SHThe vector is a load vector under the action of SH waves;
obtaining transfer functions of three directions of seismic oscillation of the offshore site by two site dynamic balance equations of a formula (21.1) and a formula (21.2); the general expression for the seismic transfer function at the offshore site is as follows:
Figure BDA0002273461360000081
in the formula ubIs the sea water bottom displacement amplitude, uoIs the displacement amplitude at the free surface of the bedrock.
Further, the step S2 specifically includes the following steps:
step S21: calculating an amplitude spectrum of the surface of the seabed soil layer;
simulating earthquake dynamic power spectrum density functions in three directions on the free surface of the bedrock by adopting a modified Tajimi-Kanai power spectrum density model, wherein the expression is as follows:
in the formula, Sbr(omega) is a seismic dynamic power spectral density function of a free surface of bedrock; | Hp(ω) | is a high-pass filter function; sg(ω) is the Tajimi-Kanai power spectral density function; omegafAnd ξfRespectively the center frequency and the damping ratio of the high-pass filter function; omegagAnd ξgThe center frequency and the damping ratio of the Tajimi-Kanai power spectral density function respectively;
from the free surface power spectrum S of the bedrockbr(omega) calculating Fourier amplitude spectrum A of free surface of bedrockr(ω), the expression is as follows:
Abr(ω)＝[4Sbr(ω)Δω]1/2(24)
where Δ ω is a frequency interval, Δ ω ═ 2 π × sampling frequency/FFT length;
after the amplitude spectrum at the free surface of the bedrock is determined, the amplitude spectrum A of the surface of the seabed soil layer of the offshore site is obtained through the offshore site transfer function H (omega)s1The expression (ω) is as follows:
As1(ω)＝|H(ω)|Abr(ω) (25)
in the formula, Abr(ω)、As1(omega) respectively represents Fourier amplitude spectrums of earthquake motion at the free surface of bedrock and the seabed soil layer of the offshore site, and H (omega) represents the earthquake motion transfer function of the offshore site obtained through calculation;
step S22: generation of [0,2 π through rand function of matlab]Uniformly distributed random phase spectrum phi1(ω)；
Step S23: inverse Fourier transform is performed based on the formula (26) to obtain a high-frequency acceleration time course a1(t)。
Step S24: for the calculated high-frequency acceleration time course a1(t) multiplying by the intensity envelope g (t) to obtain a non-stationary acceleration time course a2(t), the expression is as follows:
a2(t)＝a1(t)g(t) (27.1)
the intensity envelope consists of envelopes of a rising period, an extending period, and a falling period. The linear shape of the continuation period envelope is a horizontal line having a magnitude of 1, and the linear shapes of the rise period and fall period envelopes may be a straight line, a parabolic line, or an exponential curve.
The expressions of the envelope functions of the rising period straight line, the parabola, or the exponential curve are respectively:
straight line:
Figure BDA0002273461360000092
parabola is as follows:
Figure BDA0002273461360000093
exponential curve:
Figure BDA0002273461360000094
wherein t is a time variable; t is1Is the rise time length; c. C1Is a linear parameter of the rising time period exponential curve envelope.
The expressions for the envelope functions of the falling period straight line, the parabolic curve or the exponential curve are respectively:
straight line:
Figure BDA0002273461360000101
parabola is as follows:
Figure BDA0002273461360000102
exponential curve:
Figure BDA0002273461360000103
wherein t is a time variable; t is2The time length from 0 to the end of the extended period; c. C2A linear parameter of an envelope curve of the descending time period exponential curve; and T is the total time length of the seismic waves.
Step S25: for non-stationary acceleration time course a2(t) Fourier transform to obtain an amplitude spectrum As2(omega) and phase spectrum phi2(ω). Setting the amplitude below the critical frequency to 0 to obtain an adjusted amplitude spectrum As3(ω). Fourier inversion based on equation (27.8)Transforming to obtain high-frequency acceleration time-course component a of sea area near fault seismic oscillation3(t)。
Figure BDA0002273461360000104
Further, the step S3 specifically includes the following steps:
step S31: generating an equivalent speed pulse time course through an equivalent speed pulse model, and obtaining an equivalent acceleration time course by obtaining a derivative of the equivalent speed pulse time course:
the equivalent velocity pulse model is as follows:
v(t)＝vp·ω(t)·cos[2πfp(t-t1)]0≤t≤T (28)
in the formula, vpPeak value of the velocity pulse, fpIs the frequency of the velocity pulses, the period of which is Tp＝1/fp，-2πfPt1Is the phase angle of the velocity pulse, T is the duration of the velocity time interval, and the envelope function ω (T) of the velocity time interval is taken as follows:
Figure BDA0002273461360000105
in the formula, t0Is the peak occurrence time of the envelope function, gamma represents the peak decay rate;
obtaining the period T of the speed pulse according to a statistical formula of the speed pulse periodPAnd frequency fPThe statistical formula is as follows:
logTp＝-2.9+0.5Mw(30)
in the formula, MwThe moment magnitude is;
fP＝1/TP(31)
obtaining the peak value vP of the velocity pulse according to a statistical formula of the velocity pulse peak value, wherein the statistical formula is as follows:
log10vp＝-2.22+0.69Mw-0.58log10R (32)
in the formula, MwIs the magnitude of moment and R is the fieldThe shortest distance from ground to fault;
determining the shape parameters gamma, t from a specific pulse-type recording0And t1Taking the value of (A); a parameter TP、fP、vP、γ、t0And t1Substituting the equivalent velocity pulse model, and generating an equivalent velocity pulse time course by the equivalent velocity pulse model; obtaining an equivalent acceleration time course by derivation of the equivalent speed time course;
step S32: preliminarily obtaining a low-frequency pulse type acceleration time course:
fourier transform is carried out on the equivalent acceleration time range to obtain an amplitude spectrum A of the low-frequency acceleration time ranges4(omega) and phase spectrum phi4(ω); the amplitude spectrum of the surface of the seabed soil layer is obtained through the offshore site transfer function H (omega) as follows:
As5(ω)＝|H(ω)|As4(ω) (33)
in the formula, As4(ω)、As5(omega) respectively represents Fourier amplitude spectrums of earthquake motion at the free surface of bedrock and the seabed soil layer of the offshore site, and H (omega) represents the earthquake motion transfer function of the offshore site obtained through calculation;
finally obtaining the low-frequency pulse type acceleration time course a of seismic oscillation through the inverse Fourier transform of the formula (34)4(t)；
Figure BDA0002273461360000111
Further, the step S4 specifically includes the following steps:
step S41: regenerating a low-frequency pulse type acceleration time course;
defining that the arrival time of the low-frequency pulse acceleration peak is the same as the arrival time of the high-frequency acceleration peak; according to the definition, the low-frequency acceleration time interval is moved in parallel on the time axis, so that the peak time of the low-frequency acceleration time interval is coincident with the peak time of the high-frequency acceleration time interval;
time course a of translational acceleration on time axis4(t) obtaining the acceleration time course a of the low-frequency pulse component5(t)；
Step S42: the low-frequency pulse type acceleration time-course component a after translation is processed5(t) and the high frequency acceleration time course component a3(t) overlapping to obtain a seismic oscillation acceleration time course a (t) of the near fault in the near sea area, wherein the formula is as follows:
a(t)＝a5(t)+a3(t) (35)
compared with the prior art, the invention has the following beneficial effects:
the method can really solve the bottleneck of near fault seismic record lacking in the existing cross-sea structure seismic analysis, provides more accurate and reasonable seismic input for the seismic performance analysis of the marine structure near the seabed fault, and is beneficial to promoting and improving the seismic performance evaluation of the marine structure.
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FIG. 1 is a flow chart of an embodiment of the present invention.
Detailed Description
The invention is further explained below with reference to the drawings and the embodiments.
It should be noted that the following detailed description is exemplary and is intended to provide further explanation of the disclosure. Unless defined otherwise, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this application belongs.
It is noted that the terminology used herein is for the purpose of describing particular embodiments only and is not intended to be limiting of example embodiments according to the present application. As used herein, the singular forms "a", "an" and "the" are intended to include the plural forms as well, and it should be understood that when the terms "comprises" and/or "comprising" are used in this specification, they specify the presence of stated features, steps, operations, devices, components, and/or combinations thereof, unless the context clearly indicates otherwise.
As shown in fig. 1, the present embodiment provides a method for fitting a seabed near-fault seismic motion, including the following steps:
step S1: calculating a seismic motion transfer function H (omega) of the offshore site;
(1) calculating dynamic stiffness matrix of sea water layer
Considering that the existence of the sea water layer can influence the field amplification effect of earthquake motion, and calculating the P wave propagation speed in the sea water layer according to the parameter value of the sea water layer.
And (4) calculating the P wave incident angle of the sea water layer through the calculated phase velocity, and combining the sea water layer depth and the sea water layer damping ratio to obtain the dynamic stiffness matrix of the sea water layer.
(2) Calculating dynamic stiffness matrix of seabed soil layer
Considering that the existence of a seabed porous soil layer can influence the field amplification effect of earthquake motion, calculating the Poisson ratio of a water-containing soil body through the saturation of the seabed soil layer of an offshore field, and further calculating the P wave propagation speed of the seabed soil layer by using the Poisson ratio of the water-containing soil body;
according to the parameters of the seabed soil layer of the offshore site, calculating the propagation speeds of SH waves and SV waves in the seabed soil layer, and calculating the respective incident angles through the corresponding phase speeds;
and obtaining the dynamic stiffness matrix of the seabed soil layer by utilizing the incidence angle of the seismic waves in the seabed soil layer and the parameter value of the seabed soil layer.
(3) Calculating dynamic stiffness matrix of bedrock
Calculating the propagation velocity of seismic waves in the bedrock according to the parameter value of the bedrock in the field;
assuming the incident angle of seismic waves in bedrock, and calculating the phase velocity through the propagation velocity of the seismic waves;
and obtaining the dynamic stiffness matrix of the bedrock by utilizing the phase velocity, the seismic wave propagation velocity in the bedrock and the parameter value of the bedrock.
(4) Computing offshore site transfer functions
The dynamic stiffness matrix of the seabed field can be obtained by combining the bedrock dynamic stiffness matrix, the seabed soil layer dynamic stiffness matrix and the sea water layer dynamic stiffness matrix obtained by the above steps, and then the offshore field transfer function can be obtained by adopting a field dynamic balance equation.
Step S2: fitting the high-frequency component of the seabed seismic motion to obtain the high-frequency acceleration time-course component a of the near fault seismic motion in the near sea area3(t)；
(1) Calculating the amplitude spectrum of the surface of the seabed soil layer
And simulating the earthquake motion power spectrum at the free surface of the bedrock by adopting a power spectral density function model, and solving the Fourier amplitude spectrum at the free surface of the bedrock through a relational expression of the power spectrum and the Fourier amplitude spectrum.
And after the amplitude spectrum at the free surface of the bedrock is determined, the amplitude spectrum of the surface of the seabed soil layer is obtained through the offshore site transfer function.
(2) And calculating a random phase spectrum by using the random phase difference spectrum model.
(3) And based on the obtained amplitude spectrum and phase spectrum, obtaining a high-frequency acceleration time-course component of the seabed near fault seismic oscillation by adopting Fourier inverse transformation, and multiplying the high-frequency acceleration time-course component by an envelope function to obtain a non-stable high-frequency acceleration time-course.
Step S3: fitting the low-frequency pulse component of the ocean bottom seismic oscillation to obtain the low-frequency pulse type acceleration time course a of the seismic oscillation5(t)；
(1) And generating an equivalent speed pulse time course through an equivalent speed pulse model, and obtaining an equivalent acceleration time course by obtaining a derivative of the equivalent speed pulse time course.
(2) Considering the influence of the sea water layer on the low frequency components
And carrying out Fourier transform on the equivalent acceleration time range to obtain a phase spectrum of the low-frequency acceleration time range and an amplitude spectrum of the bedrock.
And obtaining the amplitude spectrum of the surface of the seabed soil layer based on the amplitude spectrum at the bedrock and the transfer function of the offshore site.
And obtaining the low-frequency pulse type acceleration time course through inverse Fourier transform based on the phase spectrum obtained by Fourier transform and the amplitude spectrum of the surface of the seabed soil layer.
Step S4: the high-frequency acceleration time course component a3(t) and a low-frequency pulse type acceleration time course a5And (t) superposing to obtain the seabed near fault seismic oscillation.
(1) Regenerating a low frequency pulse type acceleration time course
And translating the low-frequency acceleration time interval on a time axis to ensure that the peak time of the low-frequency acceleration time interval is coincident with the peak time of the high-frequency acceleration time interval. And regenerating a speed pulse acceleration time course to obtain a low-frequency pulse type acceleration time course component of the seabed near fault seismic oscillation.
(2) And superposing the low-frequency pulse type acceleration time-course component and the high-frequency acceleration time-course component to obtain the seabed near fault seismic oscillation acceleration simulation time-course.
In this embodiment, the step S1 specifically includes the following steps:
step S11: solving dynamic stiffness matrix [ S ] of bedrockR]；
Step S12: dynamic stiffness matrix [ S ] of sea bottom soil layerL]；
Step S13: calculating dynamic stiffness matrix [ S ] of sea water layerW]；
Step S14: dynamic stiffness matrix through sea water layerW]Dynamic stiffness matrix of seabed soil layer SL]And bedrock dynamic stiffness matrix SR]Combining to obtain a dynamic stiffness matrix of the seabed field; and determining a field dynamic balance equation through the dynamic stiffness matrix of the field, the field displacement vector and the field load vector, and solving the equation to obtain the ratio of the bottom displacement of the seabed layer to the displacement at the free surface of the bedrock, namely the offshore field transfer function H (omega).
In this embodiment, the step S11 specifically includes the following steps:
calculating the propagation velocity of seismic waves in the bedrock according to the parameter value of the bedrock in the field;
the propagation velocity of the S-wave in the bedrock is expressed as:
Figure BDA0002273461360000161
wherein G is the shear modulus of the bedrock, and rho is the density of the bedrock;
the propagation velocity of the P-wave in the bedrock is expressed as:
Figure BDA0002273461360000162
in the formula, G is the shear modulus of the bedrock, rho is the density of the bedrock, and upsilon is the Poisson ratio of the bedrock;
the S-wave velocity calculation formula considering the hysteresis damping is as follows:
in the formula, VpIs the S wave velocity, ξ is the damping ratio;
the P wave velocity calculation formula considering the hysteresis damping is as follows:
Figure BDA0002273461360000164
in the formula, VpP wave velocity, ξ damping ratio;
providing an incident angle of a P wave in the bedrock, and calculating a phase velocity through a calculation formula of the phase velocity; determining phase velocity, and sequentially deducing the incident angles of different seismic waves in other media through a calculation formula of the phase velocity;
the phase velocity is calculated as follows:
Figure BDA0002273461360000165
wherein c is the phase velocity;
Figure BDA0002273461360000171
and
Figure BDA0002273461360000172
the wave velocities of the P wave and the S wave considering the hysteresis damping are respectively considered; lxAnd mxCosine values of incident angles of the P wave and the S wave respectively; superscripts W, L and R represent the sea water layer, the seabed soil layer and the bedrock, respectively;
obtaining a dynamic stiffness matrix of the bedrock by utilizing the phase velocity, the seismic wave propagation velocity in the bedrock and the parameter value of the bedrock;
the dynamic stiffness matrix of the bedrock under the out-of-plane SH wave action is as follows:
Figure BDA0002273461360000173
in the formula, GRIs bedrock shear modulus, ξRIs the damping ratio of the bedrock, tRThe tangent value of the incident angle of the out-of-plane SH wave at the bedrock, omega is angular frequency, and c is phase velocity;
the dynamic stiffness matrix of the bedrock under the action of the P wave and the SV wave in the plane is as follows:
Figure BDA0002273461360000174
wherein,
Figure BDA0002273461360000177
Figure BDA0002273461360000178
in the formula,shear modulus of bedrock, t 'taking into account hysteresis damping'RIs tangent value, s 'of incident angle of SV wave at bedrock in plane'RThe tangent value of the incident angle of the P wave at the bedrock in the plane is shown, omega is the angular frequency, and c is the phase velocity.
In this embodiment, the step S12 specifically includes the following steps: calculating the Poisson ratio of the water-containing soil body according to the saturation of the seabed soil layer of the offshore site, wherein the Poisson ratio expression of the water-containing soil body is as follows:
Figure BDA0002273461360000182
in the formula, upsilon' and G are respectively Poisson ratio and shear modulus of a soil framework, α and M are respectively coefficients related to compressibility of soil particles and pore fluid, and porosity n and saturation S of the soil body are measuredrCalculating;
calculating the P wave propagation speed of the seabed soil layer by using the Poisson ratio of the water-containing soil body, wherein the P wave propagation speed of the seabed soil layer has the following expression:
Figure BDA0002273461360000183
wherein G is the shear modulus of the soil skeleton, α and M are coefficients related to the compressibility of soil particles and pore fluid respectively, lambda is the Lame constant of the soil skeleton, and rho is (1-n) rhos+nρfDensity of the hydrous soil body, psAnd ρfDensity of the soil particles and pore fluid, respectively;
calculating the propagation speed of S wave in the seabed soil layer according to the parameter value of the seabed soil layer in the field, wherein the S wave speed calculation formula is expressed as follows:
Figure BDA0002273461360000184
in the formula, G is the shear modulus of the soil framework, and rho is the density of the water-containing soil body;
the S-wave velocity calculation formula considering the hysteresis damping is as follows:
Figure BDA0002273461360000185
in the formula, VpIs the S wave velocity, ξ is the damping ratio;
the P wave velocity calculation formula considering the hysteresis damping is as follows:
in the formula, VpP wave velocity, ξ damping ratio;
calculating respective incidence angles through a phase velocity calculation formula; the phase velocity is calculated as follows:
Figure BDA0002273461360000192
wherein c is the phase velocity;
Figure BDA0002273461360000193
andthe wave velocities of the P wave and the S wave considering the hysteresis damping are respectively considered; lxAnd mxCosine values of incident angles of the P wave and the S wave respectively; superscripts W, L and R represent the sea water layer, the seabed soil layer and the bedrock, respectively;
and obtaining a dynamic stiffness matrix of the seabed soil layer by utilizing the incidence angle of the seismic waves in the seabed soil layer and the parameter value of the seabed soil layer. The dynamic stiffness matrix of a certain soil layer is as follows:
Figure BDA0002273461360000195
wherein each k isLRelating to angular frequency, phase velocity, seismic wave incidence angle and field parameters;
and combining the dynamic stiffness matrixes of all soil layers to obtain a dynamic stiffness matrix of the whole soil layer of the sea bottom.
In this embodiment, the step S13 specifically includes the following steps: calculating the P wave propagation speed in the sea water layer according to the parameter value of the sea water layer, wherein the P wave propagation speed expression in the sea water layer is as follows:
Figure BDA0002273461360000196
in the formula, K is the volume modulus of the seawater, and rho is the density of the seawater;
by formula (5)Calculating the incident angle of the sea water layer P wave by the calculated phase velocity, wherein c is the phase velocity;
Figure BDA0002273461360000202
and
Figure BDA0002273461360000203
the wave velocities of the P wave and the S wave considering the hysteresis damping are respectively considered; lxAnd mxCosine values of incident angles of the P wave and the S wave respectively; superscripts W, L and R represent the sea water layer, the seabed soil layer and the bedrock, respectively;
and combining the depth of the sea water layer and the damping ratio of the sea water layer to obtain a dynamic stiffness matrix of the sea water layer, which is shown in a formula (20).
Wherein each k isWAnd the parameters are related to angular frequency, phase velocity, seismic wave incidence angle, seismic wave propagation velocity, sea water layer shear modulus, sea water layer thickness, sea water layer damping ratio and the like.
In this embodiment, the step S14 specifically includes the following steps:
the field dynamic balance equation under the combined action of the P wave and the SV wave in the plane is expressed as follows:
[SP-SV]{uP-SV}＝{PP-SV} (21.1)
wherein [ S ]P-SV]Is a dynamic stiffness matrix of the in-plane direction of the seafloor field, { u }P-SVThe displacement of the field under the combined action of P wave and SV wave, and the displacement of the P wave and SV waveP-SVThe P wave and the SV wave are jointly acted to form a load vector;
the field dynamic equilibrium equation under the out-of-plane SH wave action is expressed as:
[SSH]{uSH}＝{PSH} (21.2)
wherein [ S ]SH]Is a dynamic stiffness matrix of the out-of-plane direction of the seabed ground, { u }SHThe site displacement under the action of SH wave, { P }SHThe vector is a load vector under the action of SH waves;
obtaining transfer functions of three directions of seismic oscillation of the offshore site by two site dynamic balance equations of a formula (21.1) and a formula (21.2); the general expression for the seismic transfer function at the offshore site is as follows:
Figure BDA0002273461360000211
in the formula ubIs the sea water bottom displacement amplitude, uoIs the displacement amplitude at the free surface of the bedrock.
In this embodiment, the step S2 specifically includes the following steps:
step S21: calculating an amplitude spectrum of the surface of the seabed soil layer;
simulating earthquake dynamic power spectrum density functions in three directions on the free surface of the bedrock by adopting a modified Tajimi-Kanai power spectrum density model, wherein the expression is as follows:
in the formula, Sbr(omega) is a seismic dynamic power spectral density function of a free surface of bedrock; | Hp(ω) | is a high-pass filter function; sg(ω) is the Tajimi-Kanai power spectral density function; omegafAnd ξfRespectively the center frequency and the damping ratio of the high-pass filter function; omegagAnd ξgThe center frequency and the damping ratio of the Tajimi-Kanai power spectral density function respectively;
from the free surface power spectrum S of the bedrockbr(omega) calculating Fourier amplitude spectrum A of free surface of bedrockr(ω), the expression is as follows:
Abr(ω)＝[4Sbr(ω)Δω]1/2(24)
where Δ ω is a frequency interval, Δ ω ═ 2 π × sampling frequency/FFT length;
after the amplitude spectrum at the free surface of the bedrock is determined, the amplitude spectrum A of the surface of the seabed soil layer of the offshore site is obtained through the offshore site transfer function H (omega)s1The expression (ω) is as follows:
As1(ω)＝|H(ω)|Abr(ω) (25)
in the formula, Abr(ω)、As1(omega) respectively represents Fourier amplitude spectrums of earthquake motion at the free surface of bedrock and the seabed soil layer of the offshore site, and H (omega) represents the earthquake motion transfer function of the offshore site obtained through calculation;
step S22: generation of [0,2 π through rand function of matlab]Uniformly distributed random phase spectrum phi1(ω)；
Step S23: inverse Fourier transform is performed based on the formula (26) to obtain a high-frequency acceleration time course a1(t)。
Figure BDA0002273461360000221
Step S24: for the calculated high-frequency acceleration time course a1(t) multiplying by the intensity envelope g (t) to obtain a non-stationary acceleration time course a2(t), the expression is as follows:
a2(t)＝a1(t)g(t) (27.1)
the intensity envelope consists of envelopes of a rising period, an extending period, and a falling period. The linear shape of the continuation period envelope is a horizontal line having a magnitude of 1, and the linear shapes of the rise period and fall period envelopes may be a straight line, a parabolic line, or an exponential curve.
The expressions of the envelope functions of the rising period straight line, the parabola, or the exponential curve are respectively:
straight line:
Figure BDA0002273461360000222
parabola is as follows:
Figure BDA0002273461360000223
exponential curve:
wherein t is a time variable; t is1Is the rise time length; c. C1Is a linear parameter of the rising time period exponential curve envelope.
The expressions for the envelope functions of the falling period straight line, the parabolic curve or the exponential curve are respectively:
straight line:
Figure BDA0002273461360000225
parabola is as follows:
exponential curve:
Figure BDA0002273461360000227
wherein t is a time variable; t is2The time length from 0 to the end of the extended period; c. C2A linear parameter of an envelope curve of the descending time period exponential curve; and T is the total time length of the seismic waves.
Step S25: for non-stationary acceleration time course a2(t) Fourier transform to obtain an amplitude spectrum As2(omega) and phase spectrum phi2(ω). Setting the amplitude below the critical frequency to 0 to obtain an adjusted amplitude spectrum As3(ω). Carrying out inverse Fourier transform based on the formula (27.8) to obtain a high-frequency acceleration time-course component a of the sea area near fault seismic oscillation3(t)。
Figure BDA0002273461360000231
In this embodiment, the step S3 specifically includes the following steps:
step S31: generating an equivalent speed pulse time course through an equivalent speed pulse model, and obtaining an equivalent acceleration time course by obtaining a derivative of the equivalent speed pulse time course:
an equivalent velocity pulse model is selected when the low-frequency pulse type acceleration time-course component is simulated, because the model adopts a continuous single function form to express the velocity time-course, the curve fitting of the velocity pulse and the determination of the parameters of the velocity pulse are greatly facilitated, and the equivalent velocity pulse model is as follows:
v(t)＝vp·ω(t)·cos[2πfp(t-t1)]0≤t≤T (28)
in the formula, vpPeak value of the velocity pulse, fpIs the frequency of the velocity pulses, the period of which is Tp＝1/fp，-2πfPt1Is the phase angle of the velocity pulse, T is the duration of the velocity time interval, and the envelope function ω (T) of the velocity time interval is taken as follows:
Figure BDA0002273461360000232
in the formula, t0Is the peak occurrence time of the envelope function, gamma represents the peak decay rate;
obtaining the period T of the speed pulse according to a statistical formula of the speed pulse periodPAnd frequency fPThe statistical formula is as follows:
logTp＝-2.9+0.5Mw(30)
in the formula, MwThe moment magnitude is;
fP＝1/TP(31)
obtaining the peak value v of the velocity pulse according to the statistical formula of the velocity pulse peak valuePThe statistical formula is as follows:
log10vp＝-2.22+0.69Mw-0.58log10R (32)
in the formula, MwThe moment magnitude is, and R is the shortest distance from the field to the fault;
determining the shape parameters gamma, t from a specific pulse-type recording0And t1Taking the value of (A); a parameter Tp、fp、vp、γ、t0And t1Substituting the equivalent velocity pulse model, and generating an equivalent velocity pulse time course by the equivalent velocity pulse model; obtaining an equivalent acceleration time course by derivation of the equivalent speed time course;
step S32: preliminarily obtaining a low-frequency pulse type acceleration time course:
will be equivalent toFourier transform is carried out on the acceleration time range to obtain an amplitude spectrum A of the low-frequency acceleration time ranges4(omega) and phase spectrum phi4(ω); the amplitude spectrum of the surface of the seabed soil layer is obtained through the offshore site transfer function H (omega) as follows:
As5(ω)＝|H(ω)|As4(ω) (33)
in the formula, As4(ω)、As5(omega) respectively represents Fourier amplitude spectrums of earthquake motion at the free surface of bedrock and the seabed soil layer of the offshore site, and H (omega) represents the earthquake motion transfer function of the offshore site obtained through calculation;
finally obtaining the low-frequency pulse type acceleration time course a of seismic oscillation through the inverse Fourier transform of the formula (34)4(t)；
Figure BDA0002273461360000241
In this embodiment, the step S4 specifically includes the following steps:
step S41: regenerating a low-frequency pulse type acceleration time course;
defining that the arrival time of the low-frequency pulse acceleration peak is the same as the arrival time of the high-frequency acceleration peak; according to the definition, the low-frequency acceleration time interval is moved in parallel on the time axis, so that the peak time of the low-frequency acceleration time interval is coincident with the peak time of the high-frequency acceleration time interval;
time course a of translational acceleration on time axis4(t) obtaining the acceleration time course a of the low-frequency pulse component5(t)；
Step S42: the low-frequency pulse type acceleration time-course component a after translation is processed5(t) and the high frequency acceleration time course component a3(t) overlapping to obtain a seismic oscillation acceleration time course a (t) of the near fault in the near sea area, wherein the formula is as follows:
a(t)＝a5(t)+a3(t) (35)
the above description is only a preferred embodiment of the present invention, and all equivalent changes and modifications made in accordance with the claims of the present invention should be covered by the present invention.

Claims (9)

1. A fitting method of seabed near fault seismic motion is characterized by comprising the following steps: the method comprises the following steps:
step S1: calculating a seismic motion transfer function H (omega) of the offshore site;
step S2: fitting the high-frequency component of the seabed seismic motion to obtain the high-frequency acceleration time-course component a of the near fault seismic motion in the near sea area3(t)；
Step S3: fitting the low-frequency pulse component of the ocean bottom seismic oscillation to obtain the low-frequency pulse type acceleration time course a of the seismic oscillation5(t)；
Step S4: the high-frequency acceleration time course component a3(t) and a low-frequency pulse type acceleration time course a5And (t) superposing to obtain the seabed near fault seismic oscillation.
2. The method of fitting a seafloor near fault seismic motion of claim 1, wherein: the step S1 specifically includes the following steps:
step S11: solving dynamic stiffness matrix [ S ] of bedrockR]；
Step S12: dynamic stiffness matrix [ S ] of sea bottom soil layerL]；
Step S13: calculating dynamic stiffness matrix [ S ] of sea water layerW]；
Step S14: dynamic stiffness matrix through sea water layerW]Dynamic stiffness matrix of seabed soil layer SL]And bedrock dynamic stiffness matrix SR]Combining to obtain a dynamic stiffness matrix of the seabed field; and determining a field dynamic balance equation through the dynamic stiffness matrix of the field, the field displacement vector and the field load vector, and solving the equation to obtain the ratio of the bottom displacement of the seabed layer to the displacement at the free surface of the bedrock, namely the offshore field transfer function H (omega).
3. The method of fitting a seafloor near fault seismic motion of claim 2, wherein: the step S11 specifically includes the following steps:
calculating the propagation velocity of seismic waves in the bedrock according to the parameter value of the bedrock in the field; the propagation velocity of the S-wave in the bedrock is expressed as:
Figure FDA0002273461350000011
wherein G is the shear modulus of the bedrock, and rho is the density of the bedrock;
the propagation velocity of the P-wave in the bedrock is expressed as:
in the formula, G is the shear modulus of the bedrock, rho is the density of the bedrock, and upsilon is the Poisson ratio of the bedrock;
the S-wave velocity calculation formula considering the hysteresis damping is as follows:
Figure FDA0002273461350000022
in the formula, VpIs the S wave velocity, ξ is the damping ratio;
the P wave velocity calculation formula considering the hysteresis damping is as follows:
in the formula, VpP wave velocity, ξ damping ratio;
providing an incident angle of a P wave in the bedrock, and calculating a phase velocity through a calculation formula of the phase velocity; determining phase velocity, and sequentially deducing the incident angles of different seismic waves in other media through a calculation formula of the phase velocity;
the phase velocity is calculated as follows:
Figure FDA0002273461350000024
wherein c is the phase velocity;
Figure FDA0002273461350000025
and
Figure FDA0002273461350000026
the wave velocities of the P wave and the S wave considering the hysteresis damping are respectively considered; lxAnd mxCosine values of incident angles of the P wave and the S wave respectively; superscripts W, L and R represent the sea water layer, the seabed soil layer and the bedrock, respectively;
obtaining a dynamic stiffness matrix of the bedrock by utilizing the phase velocity, the seismic wave propagation velocity in the bedrock and the parameter value of the bedrock;
the dynamic stiffness matrix of the bedrock under the out-of-plane SH wave action is as follows:
Figure FDA0002273461350000031
in the formula, GRIs bedrock shear modulus, ξRIs the damping ratio of the bedrock, tRThe tangent value of the incident angle of the out-of-plane SH wave at the bedrock, omega is angular frequency, and c is phase velocity;
the dynamic stiffness matrix of the bedrock under the action of the P wave and the SV wave in the plane is as follows:
Figure FDA0002273461350000032
wherein,
Figure FDA0002273461350000033
Figure FDA0002273461350000034
Figure FDA0002273461350000035
Figure FDA0002273461350000036
in the formula,
Figure FDA0002273461350000037
shear modulus of bedrock, t 'taking into account hysteresis damping'RIs tangent value, s 'of incident angle of SV wave at bedrock in plane'RThe tangent value of the incident angle of the P wave at the bedrock in the plane is shown, omega is the angular frequency, and c is the phase velocity.
4. The method of fitting a seafloor near fault seismic motion of claim 2, wherein: the step S12 specifically includes the following steps:
calculating the Poisson ratio of the water-containing soil body according to the saturation of the seabed soil layer of the offshore site, wherein the Poisson ratio expression of the water-containing soil body is as follows:
Figure FDA0002273461350000041
in the formula, upsilon' and G are respectively Poisson ratio and shear modulus of a soil framework, α and M are respectively coefficients related to compressibility of soil particles and pore fluid, and porosity n and saturation S of the soil body are measuredrCalculating;
calculating the P wave propagation speed of the seabed soil layer by using the Poisson ratio of the water-containing soil body, wherein the P wave propagation speed of the seabed soil layer has the following expression:
Figure FDA0002273461350000042
wherein G is the shear modulus of the soil skeleton, α and M are coefficients related to the compressibility of soil particles and pore fluid respectively, lambda is the Lame constant of the soil skeleton, and rho is (1-n) rhos+nρfDensity of the hydrous soil body, psAnd ρfDensity of the soil particles and pore fluid, respectively;
calculating the propagation speed of S wave in the seabed soil layer according to the parameter value of the seabed soil layer in the field, wherein the S wave speed calculation formula is expressed as follows:
Figure FDA0002273461350000043
in the formula, G is the shear modulus of the soil framework, and rho is the density of the water-containing soil body;
the S-wave velocity calculation formula considering the hysteresis damping is as follows:
Figure FDA0002273461350000044
in the formula, VpIs the S wave velocity, ξ is the damping ratio;
the P wave velocity calculation formula considering the hysteresis damping is as follows:
Figure FDA0002273461350000045
in the formula, VpP wave velocity, ξ damping ratio;
calculating respective incidence angles through a phase velocity calculation formula; the phase velocity is calculated as follows:
Figure FDA0002273461350000051
wherein c is the phase velocity;
Figure FDA0002273461350000052
and
Figure FDA0002273461350000053
the wave velocities of the P wave and the S wave considering the hysteresis damping are respectively considered; lxAnd mxCosine values of incident angles of the P wave and the S wave respectively; superscripts W, L and R represent the sea water layer, the seabed soil layer and the bedrock, respectively;
and obtaining a dynamic stiffness matrix of the seabed soil layer by utilizing the incidence angle of the seismic waves in the seabed soil layer and the parameter value of the seabed soil layer. The dynamic stiffness matrix of a certain soil layer is as follows:
wherein each k isLRelating to angular frequency, phase velocity, seismic wave incidence angle and field parameters;
and combining the dynamic stiffness matrixes of all soil layers to obtain a dynamic stiffness matrix of the whole soil layer of the sea bottom.
5. The method of fitting a seafloor near fault seismic motion of claim 2, wherein: the step S13 specifically includes the following steps: calculating the P wave propagation speed in the sea water layer according to the parameter value of the sea water layer, wherein the P wave propagation speed expression in the sea water layer is as follows:
in the formula, K is the volume modulus of the seawater, and rho is the density of the seawater;
by the formula
Figure FDA0002273461350000056
Calculating the incident angle of the sea water layer P wave by the calculated phase velocity, wherein c is the phase velocity;
Figure FDA0002273461350000057
and
Figure FDA0002273461350000058
the wave velocities of the P wave and the S wave considering the hysteresis damping are respectively considered; lxAnd mxCosine values of incident angles of the P wave and the S wave respectively; superscripts W, L and R represent the sea water layer, the seabed soil layer and the bedrock, respectively;
and combining the depth of the sea water layer and the damping ratio of the sea water layer to obtain a dynamic stiffness matrix of the sea water layer, which is shown in a formula (20).
Figure FDA0002273461350000061
Wherein each k isWAnd the parameters are related to angular frequency, phase velocity, seismic wave incidence angle, seismic wave propagation velocity, sea water layer shear modulus, sea water layer thickness, sea water layer damping ratio and the like.
6. The method of fitting a seafloor near fault seismic motion of claim 2, wherein: the step S14 specifically includes the following steps:
the field dynamic balance equation under the combined action of the P wave and the SV wave in the plane is expressed as follows:
[SP-SV]{uP-SV}＝{PP-SV} (21.1)
wherein [ S ]P-SV]Is a dynamic stiffness matrix of the in-plane direction of the seafloor field, { u }P-SVThe displacement of the field under the combined action of P wave and SV wave, and the displacement of the P wave and SV waveP-SVThe P wave and the SV wave are jointly acted to form a load vector;
the field dynamic equilibrium equation under the out-of-plane SH wave action is expressed as:
[SSH]{uSH}＝{PSH} (21.2)
wherein [ S ]SH]Is a dynamic stiffness matrix of the out-of-plane direction of the seabed ground, { u }SHThe site displacement under the action of SH wave, { P }SHThe vector is a load vector under the action of SH waves;
obtaining transfer functions of three directions of seismic oscillation of the offshore site by two site dynamic balance equations of a formula (21.1) and a formula (21.2); the general expression for the seismic transfer function at the offshore site is as follows:
Figure FDA0002273461350000062
in the formula ubIs the sea water bottom displacement amplitude, uoIs the displacement amplitude at the free surface of the bedrock.
7. The method of fitting a seafloor near fault seismic motion of claim 1, wherein: the step S2 specifically includes the following steps:
step S21: calculating an amplitude spectrum of the surface of the seabed soil layer;
simulating earthquake dynamic power spectrum density functions in three directions on the free surface of the bedrock by adopting a modified Tajimi-Kanai power spectrum density model, wherein the expression is as follows:
Figure FDA0002273461350000071
in the formula, Sbr(omega) is a seismic dynamic power spectral density function of a free surface of bedrock; | Hp(ω) | is a high-pass filter function; sg(ω) is the Tajimi-Kanai power spectral density function; omegafAnd ξfRespectively the center frequency and the damping ratio of the high-pass filter function; omegagAnd ξgThe center frequency and the damping ratio of the Tajimi-Kanai power spectral density function respectively;
from the free surface power spectrum S of the bedrockbr(omega) calculating Fourier amplitude spectrum A of free surface of bedrockr(ω), the expression is as follows:
Abr(ω)＝[4Sbr(ω)Δω]1/2(24)
where Δ ω is a frequency interval, Δ ω ═ 2 π × sampling frequency/FFT length;
after the amplitude spectrum at the free surface of the bedrock is determined, the amplitude spectrum A of the surface of the seabed soil layer of the offshore site is obtained through the offshore site transfer function H (omega)s1The expression (ω) is as follows:
As1(ω)＝|H(ω)|Abr(ω) (25)
in the formula, Abr(ω)、As1(omega) respectively represents Fourier amplitude spectrums of earthquake motion at the free surface of bedrock and the seabed soil layer of the offshore site, and H (omega) represents the earthquake motion transfer function of the offshore site obtained through calculation;
step S22: generation of [0,2 π through rand function of matlab]Uniformly distributed random phase spectrum phi1(ω)；
Step S23: inverse Fourier transform is performed based on the formula (26) to obtain a high-frequency acceleration time course a1(t)。
Step S24: for the calculated high-frequency acceleration time course a1(t) multiplying by the intensity envelope g (t) to obtain a non-stationary acceleration time course a2(t), the expression is as follows:
a2(t)＝a1(t)g(t) (27.1)
the intensity envelope consists of envelopes of a rising period, an extending period, and a falling period. The linear shape of the continuation period envelope is a horizontal line having a magnitude of 1, and the linear shapes of the rise period and fall period envelopes may be a straight line, a parabolic line, or an exponential curve.
The expressions of the envelope functions of the rising period straight line, the parabola, or the exponential curve are respectively:
straight line:
Figure FDA0002273461350000081
parabola is as follows:
Figure FDA0002273461350000082
exponential curve:
Figure FDA0002273461350000083
wherein t is a time variable; t is1Is the rise time length; c. C1Is a linear parameter of the rising time period exponential curve envelope.
The expressions for the envelope functions of the falling period straight line, the parabolic curve or the exponential curve are respectively:
straight line:
Figure FDA0002273461350000084
parabola is as follows:
exponential curve:
wherein t is a time variable; t is2The time length from 0 to the end of the extended period; c. C2A linear parameter of an envelope curve of the descending time period exponential curve; and T is the total time length of the seismic waves.
Step S25: for non-stationary acceleration time course a2(t) Fourier transform to obtain an amplitude spectrum As2(omega) and phase spectrum phi2(ω). Setting the amplitude below the critical frequency to 0 to obtain an adjusted amplitude spectrum As3(ω). Carrying out inverse Fourier transform based on the formula (27.8) to obtain a high-frequency acceleration time-course component a of the sea area near fault seismic oscillation3(t)。
Figure FDA0002273461350000091
8. The method of fitting a seafloor near fault seismic motion of claim 1, wherein: the step S3 specifically includes the following steps:
step S31: generating an equivalent speed pulse time course through an equivalent speed pulse model, and obtaining an equivalent acceleration time course by obtaining a derivative of the equivalent speed pulse time course:
the equivalent velocity pulse model is as follows:
v(t)＝vp·ω(t)·cos[2πfp(t-t1)]0≤t≤T (28)
in the formula, vpPeak value of the velocity pulse, fpIs the frequency of the velocity pulses, the period of which is Tp＝1/fp，-2πfPt1Is the phase angle of the velocity pulse, T is the duration of the velocity time interval, and the envelope function ω (T) of the velocity time interval is taken as follows:
Figure FDA0002273461350000092
in the formula, t0Is the peak occurrence time of the envelope function, gamma represents the peak decay rate;
obtaining the period T of the speed pulse according to a statistical formula of the speed pulse periodPAnd frequency fPThe statistical formula is as follows:
logTp＝-2.9+0.5Mw(30)
in the formula, MwThe moment magnitude is;
fP＝1/TP(31)
obtaining the peak value v of the velocity pulse according to the statistical formula of the velocity pulse peak valuePThe statistical formula is as follows:
log10vp＝-2.22+0.69Mw-0.58log10R (32)
in the formula, MwThe moment magnitude is, and R is the shortest distance from the field to the fault;
determining the shape parameters gamma, t from a specific pulse-type recording0And t1Taking the value of (A); a parameter Tp、fp、vp、γ、t0And t1Substituting the equivalent velocity pulse model, and generating an equivalent velocity pulse time course by the equivalent velocity pulse model; obtaining an equivalent acceleration time course by derivation of the equivalent speed time course;
step S32: preliminarily obtaining a low-frequency pulse type acceleration time course:
fourier transform is carried out on the equivalent acceleration time range to obtain an amplitude spectrum A of the low-frequency acceleration time ranges4(omega) and phase spectrum phi4(ω); the amplitude spectrum of the surface of the seabed soil layer is obtained through the offshore site transfer function H (omega) as follows:
As5(ω)＝|H(ω)|As4(ω) (33)
in the formula, As4(ω)、As5(omega) respectively represents Fourier amplitude spectrums of earthquake motion at the free surface of bedrock and the seabed soil layer of the offshore site, and H (omega) represents the earthquake motion transfer function of the offshore site obtained through calculation;
finally obtaining the low-frequency pulse type acceleration time course a of seismic oscillation through the inverse Fourier transform of the formula (34)4(t)；
Figure FDA0002273461350000101
9. The method of fitting a seafloor near fault seismic motion of claim 1, wherein: the step S4 specifically includes the following steps:
step S41: regenerating a low-frequency pulse type acceleration time course;
defining that the arrival time of the low-frequency pulse acceleration peak is the same as the arrival time of the high-frequency acceleration peak; according to the definition, the low-frequency acceleration time interval is moved in parallel on the time axis, so that the peak time of the low-frequency acceleration time interval is coincident with the peak time of the high-frequency acceleration time interval;
time course a of translational acceleration on time axis4(t) obtaining the acceleration time course a of the low-frequency pulse component5(t)；
Step S42: the low-frequency pulse type acceleration time-course component a after translation is processed5(t) and the high frequency acceleration time course component a3(t) overlapping to obtain a seismic oscillation acceleration time course a (t) of the near fault in the near sea area, wherein the formula is as follows:
a(t)＝a5(t)+a3(t) (35)
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